Quadratic twists of the elliptic curve 14a2

In this paper, we consider a quadratic twist family of the elliptic curve 14a2. Using the classical 2-descent method, we prove that all curves in this family have 2-Selmer group (Z/2Z)². Assume the finiteness of the Shafarevich-Tate group, then these curves have rank 1, Selmer Z₂-corank 1, and trivial 2-primary part of the Shafarevich-Tate group. According to the Gross-Zagier formula and the Tunnell-Saito theorem, we raise the question of how to prove the non-triviality of certain explicit Heegner points defined on Shimura curves associated to the quaternion algebra precisely ramified at 2 and 7.